Mixing
plane given with a line on it -> build a rectangle in the plane with the line as the diagonal of the...
$begingroup$
(Everything's in 3D)
Firstly a plane is given. On this plane I'm drawing a line. The beginning and ending point of the line is given therefore.
What I have to do is to use the line as the diagonal of the rectangle to create the rectangle. For this I just need the other two missing points which are also in the very same plane I've got.
How can I determine the missing two points?
In 2D for example if you have like point B (6|4) and C (1|2) then you can conclude that A is on (1|4) and D is on (6|2).
But I struggle to find a method/algorithm to do so in a 3D world.
PS: If I used the wrong tag please tell me another suggestion, thx!
geometry
asked Jan 16 at 7:52
user9268398user9268398
12
$endgroup$
add a comment |
$begingroup$
(Everything's in 3D)
Firstly a plane is given. On this plane I'm drawing a line. The beginning and ending point of the line is given therefore.
What I have to do is to use the line as the diagonal of the rectangle to create the rectangle. For this I just need the other two missing points which are also in the very same plane I've got.
How can I determine the missing two points?
In 2D for example if you have like point B (6|4) and C (1|2) then you can conclude that A is on (1|4) and D is on (6|2).
But I struggle to find a method/algorithm to do so in a 3D world.
PS: If I used the wrong tag please tell me another suggestion, thx!
geometry
asked Jan 16 at 7:52
user9268398user9268398
12
$endgroup$
1
$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37
add a comment |
$begingroup$
(Everything's in 3D)
Firstly a plane is given. On this plane I'm drawing a line. The beginning and ending point of the line is given therefore.
What I have to do is to use the line as the diagonal of the rectangle to create the rectangle. For this I just need the other two missing points which are also in the very same plane I've got.
How can I determine the missing two points?
In 2D for example if you have like point B (6|4) and C (1|2) then you can conclude that A is on (1|4) and D is on (6|2).
But I struggle to find a method/algorithm to do so in a 3D world.
PS: If I used the wrong tag please tell me another suggestion, thx!
geometry
asked Jan 16 at 7:52
user9268398user9268398
12
$endgroup$
(Everything's in 3D)
Firstly a plane is given. On this plane I'm drawing a line. The beginning and ending point of the line is given therefore.
What I have to do is to use the line as the diagonal of the rectangle to create the rectangle. For this I just need the other two missing points which are also in the very same plane I've got.
How can I determine the missing two points?
In 2D for example if you have like point B (6|4) and C (1|2) then you can conclude that A is on (1|4) and D is on (6|2).
But I struggle to find a method/algorithm to do so in a 3D world.
PS: If I used the wrong tag please tell me another suggestion, thx!
geometry
geometry
asked Jan 16 at 7:52
user9268398user9268398
12
asked Jan 16 at 7:52
user9268398user9268398
12
edited Jan 16 at 7:55
user9268398
asked Jan 16 at 7:52
user9268398user9268398
12
asked Jan 16 at 7:52
user9268398user9268398
12
asked Jan 16 at 7:52
user9268398user9268398
12
12
1
$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37
add a comment |
1
$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37
1
1
$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37
add a comment |
1 Answer
1
active
oldest
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$begingroup$
In your 2-D example, you can only reach that conclusion about the other vertices of the rectangle because you’re making the unstated assumption that the sides of the rectangle are parallel to the coordinate axes. If you remove that assumption, then there’s an infinite number of rectangles with that diagonal.
You’re in the same situation in 3-D. There’s no obvious counterpart to “parallel to the axes” for an arbitrary plane, so you’ll need to come up with additional constraints to determine the rectangle uniquely.
answered Jan 17 at 0:24
amdamd
30.5k21050
$endgroup$
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
add a comment |
Your Answer
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1 Answer
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$begingroup$
In your 2-D example, you can only reach that conclusion about the other vertices of the rectangle because you’re making the unstated assumption that the sides of the rectangle are parallel to the coordinate axes. If you remove that assumption, then there’s an infinite number of rectangles with that diagonal.
You’re in the same situation in 3-D. There’s no obvious counterpart to “parallel to the axes” for an arbitrary plane, so you’ll need to come up with additional constraints to determine the rectangle uniquely.
answered Jan 17 at 0:24
amdamd
30.5k21050
$endgroup$
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
add a comment |
$begingroup$
In your 2-D example, you can only reach that conclusion about the other vertices of the rectangle because you’re making the unstated assumption that the sides of the rectangle are parallel to the coordinate axes. If you remove that assumption, then there’s an infinite number of rectangles with that diagonal.
You’re in the same situation in 3-D. There’s no obvious counterpart to “parallel to the axes” for an arbitrary plane, so you’ll need to come up with additional constraints to determine the rectangle uniquely.
answered Jan 17 at 0:24
amdamd
30.5k21050
$endgroup$
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
add a comment |
$begingroup$
In your 2-D example, you can only reach that conclusion about the other vertices of the rectangle because you’re making the unstated assumption that the sides of the rectangle are parallel to the coordinate axes. If you remove that assumption, then there’s an infinite number of rectangles with that diagonal.
You’re in the same situation in 3-D. There’s no obvious counterpart to “parallel to the axes” for an arbitrary plane, so you’ll need to come up with additional constraints to determine the rectangle uniquely.
answered Jan 17 at 0:24
amdamd
30.5k21050
$endgroup$
In your 2-D example, you can only reach that conclusion about the other vertices of the rectangle because you’re making the unstated assumption that the sides of the rectangle are parallel to the coordinate axes. If you remove that assumption, then there’s an infinite number of rectangles with that diagonal.
You’re in the same situation in 3-D. There’s no obvious counterpart to “parallel to the axes” for an arbitrary plane, so you’ll need to come up with additional constraints to determine the rectangle uniquely.
answered Jan 17 at 0:24
amdamd
30.5k21050
answered Jan 17 at 0:24
amdamd
30.5k21050
answered Jan 17 at 0:24
amdamd
30.5k21050
answered Jan 17 at 0:24
amdamd
30.5k21050
30.5k21050
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
add a comment |
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
thanks finally I do understand it.
$endgroup$
– user9268398
Jan 17 at 1:03
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
so if I make the rectangle parallel to an extra plane then I could build it up right? in this case the extra plane should be perpendicular to the plane given in the first place.
$endgroup$
– user9268398
Jan 17 at 1:08
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
Equivalently, pick some line on the plane to define the directions of the sides of the rectangle.
$endgroup$
– amd
Jan 17 at 1:10
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
I'm sorry but actually I don't get how I can define directions by that.
$endgroup$
– user9268398
Jan 17 at 1:40
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
$begingroup$
Call that line the “$x$-axis” on the plane. The “$y$-axis” is orthogonal to it, so now you’ve got the directions of the rectangle sides.
$endgroup$
– amd
Jan 17 at 17:45
add a comment |
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$begingroup$
Note that infinitely many rectangles can be formed of two given points . In your case , the points you found are two of many . To form a unique rectangle , one needs atleast 3 points .
$endgroup$
– Rahuboy
Jan 16 at 14:49
$begingroup$
The corners of the rectangle are on a circle of which the diagonal is the diameter. The circle is of course on the plane.
$endgroup$
– Moti
Jan 16 at 19:07
$begingroup$
Is there still infinite possible rectangle if I fix both points to a defined plane? I mean shouldn't it then behave exactly as in 2D?
$endgroup$
– user9268398
Jan 17 at 0:14
$begingroup$
How is the line defined? Is it as two 3D points that happen to be on the line, or are the two points given on some local coordinates on the plane?
$endgroup$
– ja72
Jan 17 at 0:37